In her thesis, Mirzakhani showed that the number of simple closed geodesics of length $\leq L$ on a closed, connected, oriented hyperbolic surface X of genus g is asymptotic to $L^{6g-6}$ times a constant depending on the geometry of X. In this survey, we give a detailed account of Mirzakhani’s proof of this result aimed at non-experts. We draw inspiration from classic primitive lattice point counting results in homogeneous dynamics. The focus is on understanding how the general principles that drive the proof in the case of lattices also apply in the setting of hyperbolic surfaces.

In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of random and almost periodic potentials.

We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space of a closed orientable surface of genus at least 2, then we explain the construction for Hitchin components and finally we treat the general case. This paper surveys results of Bridgeman, Canary, Labourie and Sambarino, The pressure metric for Anosov representations, and discusses questions and open problems which arise.

We survey the theory of quasi-periodic Schrödinger-type operators, focusing on the advances made since the early 2000s by adopting a dynamical systems point of view.

The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group$\mathrm {SL}(d,{\mathbb {R}})$ and we explain how classical tools and new techniques enter the proof: the Auslander projection theorem, the Bruhat decomposition, the Mahler compactness criterion, the Borel density theorem, the Borel–Harish-Chandra finiteness theorem, the Howe–Moore mixing theorem, the Dani–Margulis recurrence theorem, the Raghunathan–Venkataramana finite-index subgroup theorem and so on.

This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

Ergodic optimization is the study of problems relating to maximizing orbits and invariant measures, and maximum ergodic averages. An orbit of a dynamical system is called $f$-maximizing if the time average of the real-valued function $f$ along the orbit is larger than along all other orbits, and an invariant probability measure is called $f$-maximizing if it gives $f$ a larger space average than any other invariant probability measure. In this paper, we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.

Kolmogorov–Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.

Partial hyperbolicity appeared in the 1960s as a natural generalization of hyperbolicity. In the last 20 years, there has been great activity in this area. Here we survey the state of the art in some related topics, focusing especially on partial hyperbolicity in dimension three. The reason for this is not only that it is the smallest dimension in which non-degenerate partial hyperbolicity can occur, but also that the topology of $3$-manifolds influences the dynamics in revealing ways.